Rachel is in Vegas, and notices a hustler's stall outside one of the casinos.

Hustler: "There are two normal pennies in here, an I'm offering 2-1 on you getting two heads when tossed."
Rachel: "I refuse those odds. They're biased in your favour."
Hustler: "There are three different results, two heads, two tails, or a head and a tail. So 2-1 is fair."
Rachel: "Right then. Can you offer me 2-1 on a head and a tail, then ? "

(Hustler thinks and diverts the question)

Hustler: "All right. There are ten normal pennies in here, and I'll give you the same 2-1 odds for getting 5 heads out of 10. I mean, half of 10 is 5, so I should only be offering you evens."
Rachel: "No. The chance of getting 5 heads out of 10 is even lower than getting 2 heads out of 2."

Based on the rules of probability every time you toss a coin it's an independent event. You have a 1 in 2 chance of hitting a head or tail each time. So to hit 2 heads it would be 1/2*1/2 equalling 1/4 which is odds of 4-1.
For the 5 heads from ten coins the best chance is 1/2*1/2*1/2*1/2*1/2 equalling 1/32 or 32-1 in terms of odds. However, that relies on hitting 5 heads in the first 5 goes. If that didn't happen the odds would go up in this order: 64-1; 128-1; 256-1; 512-1; 1024-1.

My understanding is that a 1 in 32 chance would be classed as 31-1 in terms of fair odds.

Do we mean exactly 5 heads or at least 5? For exactly 5 then the probability would be 0.5 to the power of 10, multiplied by the number of different ways this can be achieved eg the heads might be tosses 1 2 3 4 5 or tosses 2 3 6 7 9 etc etc. At school we used Pascal's triangle to arrive at this number. Can't be arsed checking it now but that's the principle anyway.

The relevant number from the triangle is 252, so the probability of exactly 5 heads out of 10 is 24.61%.

I therefore assume this is what the original question was getting at, as the probability of at least 5 is just over 62%.

Rachel was correct in both cases :

The prob. of 2 heads out of 2 is 1/4, so the true odds would be 3-1 against, not 2-1.
As a head and a tail can be made in 2 ways out of a possible 4, the true odds would be evens,
which is why the hustler changed his mind when Rachel suggested 2-1 for a head and a tail.

As for the case of 5 heads in 10, the Pascal's triangle entry is 252, so the prob. of exactly 5 heads in 10 is 252/1024 or 63/256, which is very slightly less than 1 in 4. The fair odds would be 193 to 63, or about 3.06 to 1.

This is an example of binomial distribution, for all statisticians among you.